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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ofcc | Structured version Visualization version GIF version |
Description: Left operation by a constant on a mixed operation with a constant. (Contributed by Thierry Arnoux, 31-Jan-2017.) |
Ref | Expression |
---|---|
ofcc.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
ofcc.2 | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
ofcc.3 | ⊢ (𝜑 → 𝐶 ∈ 𝑋) |
Ref | Expression |
---|---|
ofcc | ⊢ (𝜑 → ((𝐴 × {𝐵})∘𝑓/𝑐𝑅𝐶) = (𝐴 × {(𝐵𝑅𝐶)})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ofcc.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
2 | fnconstg 6393 | . . . 4 ⊢ (𝐵 ∈ 𝑊 → (𝐴 × {𝐵}) Fn 𝐴) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → (𝐴 × {𝐵}) Fn 𝐴) |
4 | ofcc.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
5 | ofcc.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑋) | |
6 | fvconst2g 6789 | . . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝑥 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝑥) = 𝐵) | |
7 | 1, 6 | sylan 572 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝑥) = 𝐵) |
8 | 3, 4, 5, 7 | ofcfval 31033 | . 2 ⊢ (𝜑 → ((𝐴 × {𝐵})∘𝑓/𝑐𝑅𝐶) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶))) |
9 | fconstmpt 5460 | . 2 ⊢ (𝐴 × {(𝐵𝑅𝐶)}) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶)) | |
10 | 8, 9 | syl6eqr 2825 | 1 ⊢ (𝜑 → ((𝐴 × {𝐵})∘𝑓/𝑐𝑅𝐶) = (𝐴 × {(𝐵𝑅𝐶)})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1508 ∈ wcel 2051 {csn 4435 ↦ cmpt 5004 × cxp 5401 Fn wfn 6180 ‘cfv 6185 (class class class)co 6974 ∘𝑓/𝑐cofc 31030 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pr 5182 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-ral 3086 df-rex 3087 df-reu 3088 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-nul 4173 df-if 4345 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4709 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-id 5308 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-ov 6977 df-oprab 6978 df-mpo 6979 df-ofc 31031 |
This theorem is referenced by: (None) |
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